In this paper, we propose a novel concept for engineered molecular communication (MC) systems inspired by animal olfaction. We focus on a multi-user scenario where several transmitters wish to communicate with a central receiver. We assume that each transmitter employs a unique mixture of different types of signaling molecules to represent its message and the receiver is equipped with an array comprising $R$ different types of receptors in order to detect the emitted molecule mixtures. The design of an MC system based on \textit{orthogonal} molecule-receptor pairs implies that the hardware complexity of the receiver linearly scales with the number of signaling molecule types $Q$ (i.e., $R=Q$). Natural olfaction systems avoid such high complexity by employing arrays of \textit{cross-reactive} receptors, where each type of molecule activates multiple types of receptors and each type of receptor is predominantly activated by multiple types of molecules albeit with different activation strengths. For instance, the human olfactory system is believed to discriminate several thousands of chemicals using only a few hundred receptor types, i.e., $Q\gg R$. Motivated by this observation, we first develop an end-to-end MC channel model that accounts for the key properties of olfaction. Subsequently, we present the proposed transmitter and receiver designs. In particular, given a set of signaling molecules, we develop algorithms that allocate molecules to different transmitters and optimize the mixture alphabet for communication. Moreover, we formulate the molecule mixture recovery as a convex compressive sensing problem which can be efficiently solved via available numerical solvers.